Download Magnetic quenching of turbulent diffusivity - Andrés Muñoz

The Astrophysical Journal Letters, 727:L23 (6pp), 2011 January 20
C 2011.
doi:10.1088/2041-8205/727/1/L23
The American Astronomical Society. All rights reserved. Printed in the U.S.A.
MAGNETIC QUENCHING OF TURBULENT DIFFUSIVITY: RECONCILING MIXING-LENGTH THEORY
ESTIMATES WITH KINEMATIC DYNAMO MODELS OF THE SOLAR CYCLE
Andrés Muñoz-Jaramillo1,2 , Dibyendu Nandy3 , and Petrus C. H. Martens1,2
1
3
Department of Physics, Montana State University, Bozeman, MT 59717, USA; [email protected]
2 Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA; [email protected]
Indian Institute for Science Education and Research, Kolkata, Mohampur 741252, West Bengal, India; [email protected]
Received 2010 May 4; accepted 2010 December 10; published 2010 December 30
ABSTRACT
The turbulent magnetic diffusivity in the solar convection zone is one of the most poorly constrained ingredients of
mean-field dynamo models. This lack of constraint has previously led to controversy regarding the most appropriate
set of parameters, as different assumptions on the value of turbulent diffusivity lead to radically different solar cycle
predictions. Typically, the dynamo community uses double-step diffusivity profiles characterized by low values
of diffusivity in the bulk of the convection zone. However, these low diffusivity values are not consistent with
theoretical estimates based on mixing-length theory, which suggest much higher values for turbulent diffusivity. To
make matters worse, kinematic dynamo simulations cannot yield sustainable magnetic cycles using these theoretical
estimates. In this work, we show that magnetic cycles become viable if we combine the theoretically estimated
diffusivity profile with magnetic quenching of the diffusivity. Furthermore, we find that the main features of this
solution can be reproduced by a dynamo simulation using a prescribed (kinematic) diffusivity profile that is based
on the spatiotemporal geometric average of the dynamically quenched diffusivity. This bridges the gap between
dynamically quenched and kinematic dynamo models, supporting their usage as viable tools for understanding the
solar magnetic cycle.
Key words: Sun: activity – Sun: dynamo – Sun: interior
Online-only material: color figures
we find that the turbulent diffusivity coefficient becomes (Moffat
1978)
τ
η = v 2 ,
(1)
3
where τ is the eddy correlation time and v corresponds to the
turbulent velocity field. In order to make an order of magnitude
estimate we turn to MLT, which although not perfect has been
found to be in general agreement with numerical simulations
of turbulent convection (Chan & Sofia 1987; Abbett et al.
1997). More specifically we use the Solar Model S (ChristensenDalsgaard et al. 1996), which is a comprehensive solar interior
model used by GONG in all their helioseismic calculations.
Among other quantities, this model estimates the mixing-length
parameter αp , the convective velocity v for different radii, and
the necessary variables to calculate the pressure scale height Hp .
In terms of those quantities the diffusivity becomes
1. INTRODUCTION
The solar magnetic cycle involves the recycling of the
toroidal and poloidal components of the magnetic field, which
are generated at spatially segregated source layers that must
communicate with each other (see, e.g., Wilmot-Smith et al.
2006; Charbonneau 2010). This communication is mediated
via magnetic flux transport, which in most kinematic solar
dynamo models is achieved through diffusive and advective
(i.e., by meridional circulation) transport of magnetic fields.
The relative strength of turbulent diffusion and meridional
circulation determines the regime in which the solar cycle
operates, and this has far reaching implications for cycle
memory and solar cycle predictions (Yeates et al. 2008; Nandy
2010). As shown in Yeates et al. (2008), different assumptions
on the strength of turbulent diffusivity in the bulk of the solar
convection zone (SCZ) lead to different predictions for the solar
cycle (Dikpati et al. 2006; Choudhuri et al. 2007). Previously,
this lack of constraint has led to controversy regarding what
value of turbulent diffusivity is more appropriate and yields
better solar-like solutions (Nandy & Choudhuri 2002; Dikpati
et al. 2002; Chatterjee et al. 2004; Dikpati et al. 2005; Choudhuri
et al. 2005). Currently, most dynamo modelers use double-step
diffusivity profiles that are somewhat ad hoc and different from
one another (see Figure 1; Rempel 2006a; Dikpati & Gilman
2007; Guerrero & de Gouveia Dal Pino 2007; Jouve & Brun
2007). There is, however, a way of theoretically estimating the
radial dependence of magnetic diffusivity based on mixinglength theory (MLT; Prandtl 1925).
η∼
1
αp Hp v,
3
(2)
which we plot in Figure 1 (solid black line) and show how it
compares to commonly used diffusivity profiles.
3. THE PROBLEM AND A POSSIBLE SOLUTION
It is evident that there is a major discrepancy between the
theoretical estimate and the typical values used inside the
convection zone (around two orders of magnitude difference).
This creates a problem that is aggravated by the fact that dynamo
models simply cannot operate under very strong diffusivities as
suggested by MLT. A possible solution for this inconsistency
resides in the back-reaction that strong magnetic fields have on
velocity fields, which results in a suppression of turbulence and
thus of turbulent magnetic diffusivity (Roberts & Soward 1975).
2. ORDER OF MAGNITUDE ESTIMATION
Going back to the derivation of the mean-field dynamo
equations (after using the first-order smoothing approximation),
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The Astrophysical Journal Letters, 727:L23 (6pp), 2011 January 20
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Muñoz-Jaramillo, Nandy, & Martens
2. Is there any justification for using kinematic diffusivity
profiles vis-á-vis the effective turbulent diffusivity after
taking quenching into account?
Radial Dependence of the Turbulent Magnetic Diffusivity
10
13
10
4. THE KINEMATIC MEAN-FIELD DYNAMO MODEL
Our model is based on the axisymmetric dynamo equations:
∂A 1
1
2
+ [vp · ∇(sA)] = η ∇ − 2 A
∂t
s
s
∂B
B
1
2
+ s vp · ∇
+ (∇ · vp )B = η ∇ − 2 B
∂t
s
s
1 ∂η ∂(sB)
1 ∂η ∂(sB)
+ s ∇ × (Aêφ ) · ∇ Ω +
+ 2
, (3)
s ∂r ∂r
sr ∂θ ∂θ
12
η (cm2 /s)
10
11
10
10
10
9
10
where A is the φ-component of the potential vector (from
which Br and Bθ can be obtained), B is the toroidal field
(Bφ ), vp is the meridional flow, Ω is the differential rotation,
η is the turbulent magnetic diffusivity, and s = r sin(θ ). In
order to integrate these equations, we need to prescribe four
ingredients: meridional flow, differential rotation, poloidal field
regeneration mechanism, and turbulent magnetic diffusivity.
We use the same meridional flow profile defined by MuñozJaramillo et al. (2009), which better captures the features present
in helioseismic data. Our flow profile has a penetration depth
of 0.675 R and a top speed of 15.0 m s−1 . For the differential
rotation we use the analytical form of Charbonneau et al. (1999),
with a tachocline centered at 0.7 R and whose thickness is
0.05 R . For the poloidal field regeneration mechanism, we
use the improved ring-duplet algorithm described by MuñozJaramillo et al. (2010) and Nandy et al. (2010), but using a value
of K0 = 3900, in order to ensure supercriticality. As MuñozJaramillo et al. (2010), we restrict active region (AR) emergence
to latitudes between 45◦ N and 0◦ S since observations show that
active latitudes are located only within this range (Tang et al.
1984; Wang & Sheeley 1989). Specifics about our treatment of
the turbulent diffusivity and numerical methods are described
below. More details regarding kinematic dynamo models can
be found in a review by Charbonneau (2010) and references
therein.
8
10
0.55
0.6
0.65
0.7
0.75 0.8
r/Rs
0.85
0.9
0.95
MLT and ModelS of Christensen−Dalsgaard 1996
Dikpati & Gilman 2007
Nandy & Choudhuri 2002
Guerrero & de Gouveia Dal Pino 2007
Rempel 2006
Jouve & Brun 2007
Figure 1. Different diffusivity profiles used in kinematic dynamo simulations.
The solid black line corresponds to an estimate of turbulent diffusivity obtained
by combining mixing-length theory (MLT) and the Solar Model S. The fact that
viable solutions can be obtained with such a varied array of profiles has led
to debates regarding which profile is more appropriate. Nevertheless, it is well
known that kinematic dynamo simulations cannot yield viable solutions using
the MLT estimate.
(A color version of this figure is available in the online journal.)
Magnetic “quenching” of the turbulent diffusivity has been
studied before in different contexts by dynamo modelers:
Rüdiger et al. (1994) studied quenching in the context of stellar
and galactic geometries finding that the effect of quenching
is more important in steady dynamos than in oscillating ones;
Tobias (1996) studied quenching in the context of interface
dynamos finding that quenching allows for the restriction of the
magnetic field to a narrow layer in the interface between the
convection and radiative zones and its amplification to superequipartition values; Gilman & Rempel (2005) studied how
quenching can contribute to the generation of strong superequipartition fields at the bottom of the convection zone, even if
one accounts for the magnetic feedback on differential rotation;
and Guerrero et al. (2009) studied the combined effect of
diffusivity quenching and poloidal source quenching finding
that although diffusivity quenching leads to magnetic field
amplification, it also produces solutions which are farther from
observations than those of standard kinematic simulations.
In this study, we continue the work in progress presented
by Muñoz-Jaramillo et al. (2008) and apply for the first time
diffusivity quenching to actual MLT values thanks to current improvements in computational techniques (Hochbruck &
Lubich 1997; Hochbruck et al. 1998; Muñoz-Jaramillo et al.
2009).
We focus on two very specific issues that have never been
addressed before.
4.1. Turbulent Magnetic Diffusivity and Diffusivity Quenching
As opposed to previous works in which diffusivity is
quenched instantly (Tobias 1996; Gilman & Rempel 2005;
Muñoz-Jaramillo et al. 2008; Guerrero et al. 2009), we introduce an additional state variable ηmq , in order to study the effect
of magnetic quenching on dynamo models, governed by the
following differential equation:
∂ηmq
1
ηMLT (r)
=
− ηmq (r, θ, t) .
(4)
∂t
τ 1 + B2 (r, θ, t)/B02
This treatment allows for an efficient implementation in the
context of our numerical scheme (described in Muñoz-Jaramillo
et al. 2009), but has at its core the quenching form used by
Gilman & Rempel (2005), Muñoz-Jaramillo et al. (2008), and
Guerrero et al. (2009), which depends on the magnetic energy in
reference to its equipartition counterpart. This has been found
to describe the suppression of turbulent diffusivity by MHD
simulations (Yousef et al. 2003).
In a steady state, ηmq corresponds to the MLT estimated
diffusivity ηMLT (r) quenched in such a way that the diffusivity
1. Does introducing magnetic quenching of the diffusivity
allow the dynamo to reach viable oscillatory solutions if
one starts from an MLT estimated profile?
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The Astrophysical Journal Letters, 727:L23 (6pp), 2011 January 20
Muñoz-Jaramillo, Nandy, & Martens
unbound magnetic field growth. By putting a limit on how small
can the diffusivity become, we successfully avoid this type
of computational instability. Note that Guerrero et al. (2009)
find pervasive small-scale structures for diffusivities lower than
109 cm2 s−1 which we avoid by placing this lower limit.
Turbulent Magnetic Diffusivity
14
10
13
10
12
η (cm 2 /s)
10
5. DYNAMO SIMULATIONS
11
10
We use the SD-Exp4 code (see the Appendix of MuñozJaramillo et al. 2009) to solve the dynamo equations
(Equation (3)). Our computational domain comprises the SCZ
and upper layer of the solar radiative zone in the northern hemisphere (0.55 R r R and 0 θ π ). In order to
approximate the spatial differential operators with finite differences we use a uniform grid (in radius and co-latitude), with a
resolution of 400 × 400 grid points.
Our boundary conditions assume that the toroidal and radial
magnetic fields are anti-symmetric and the co-latitudinal field is
symmetric across the equator (∂A/∂θ |θ=π/2 = 0; B(r, π/2) =
0), that the plasma below the lower boundary is a perfect
conductor (A(r = 0.55 R , θ ) = 0; ∂(rB)/∂r|r=0.55 R = 0),
that the magnetic field is axisymmetric (A(r, θ = 0) = 0;
B(r, θ = 0) = 0), and that the field at the surface is radial
(∂(rA)/∂r|r=R = 0; B(r = R , θ ) = 0). Our initial conditions
consist of a large toroidal belt and no poloidal component. After
setting up the problem we let the magnetic field evolve for 200
years allowing the dynamo to reach a stable cycle.
10
10
9
10
Eta
Fit
EtaMLT
8
10
EtaMin
0.6
0.7
r/R
0.8
0.9
1
s
Figure 2. Fit (solid line) of diffusivity as a function of radius to the mixinglength theory estimate (circles). As part of our definition of effective diffusivity
we put a limit on how much can the diffusivity be quenched. This minimum
diffusivity has a radial dependence that is shown as a red dashed line.
(A color version of this figure is available in the online journal.)
is halved for a magnetic field of amplitude B0 = 6700 Gauss
(G). This value corresponds to the average equipartition field
strength inside the SCZ calculated using the Solar Model S.
The average eddy turnover time in the SCZ calculated using
Solar Model S corresponds to τ = 30 days. Therefore, the
dynamical effect of a magnetic field will be felt locally over this
characteristic timescale. For comparison, we also performed
simulations using τ = 20 days and τ = 10 days and did not
find any major differences in the results. This is to be expected
because this timescale is negligible compared to the dynamo
cycle period.
We make a fit of ηMLT (r) using the following analytical profile
(see Figure 2):
r − r1
η1 − η0
η(r) = η0 +
1 + erf
2
d1
r − r2
η2 − η1 − η0
1 + erf
,
(5)
+
2
d2
6. RESULTS AND DISCUSSION
The first important result is the existence of a uniform cycle
in dynamic equilibrium. The presence of diffusivity quenching
allows the dynamo to become viable in a regime in which
kinematic dynamo models cannot operate, thanks to the creation
of pockets of relatively low magnetic diffusivity (where long
lived magnetic structures can exist). This can be clearly seen
in Figure 3, which shows snapshots of the effective turbulent
diffusivity and the toroidal and poloidal components of the
magnetic field at different moments during the sunspot cycle
(half a magnetic cycle). As expected the turbulent diffusivity
is strongly suppressed by the magnetic field (especially by the
toroidal component), increasing the diffusive timescale to the
point where diffusion and advection become equally important
for flux transport dynamics. This slowdown of the diffusive
process is crucial for the survival of the magnetic cycle since
it gives differential rotation more time to amplify the weak
poloidal components of the magnetic field into strong toroidal
belts, while providing them a measure of isolation from the top
(r = R ) and polar (θ = 0) boundary conditions (B = 0).
where η0 = 108 cm2 s−1 corresponds to the diffusivity at the
bottom of the computational domain; η1 = 1.4 × 1013 cm2 s−1
and η2 = 1010 cm2 s−1 control the diffusivity in the convection
zone; r1 = 0.71 R , d1 = 0.015 R , r2 = 0.96 R , and
d2 = 0.09 R characterize the transitions from one value of
diffusivity to another. With this in mind, we define the effective
diffusivity at any given point as
ηeff (r, θ, t) = ηmin (r) + ηmq (r, θ, t),
6.1. Spatiotemporal Averages of the Effective Diffusivity
Given that we ultimately want to understand how adequate
current kinematic diffusivity profiles are and whether they are
plausible representations of physical reality consistent with
MLT, we seek to explore the connection between kinematic
profiles and the dynamically quenched diffusivity. Because of
this, the next natural step is to find spatiotemporal averages of
the effective diffusivity. For this purpose we use the generalized
mean:
⎛
⎞1/p
1
ηav (ri ) = Mp = ⎝
ηeff (ri , θj , tn )p ⎠ , (8)
Nθ Nt j n
(6)
with the minimum magnetic diffusivity ηmin (r) given by the
following analytical profile (see Figure 2):
r − rcz
ηcz − η0
ηmin (r) = η0 +
,
(7)
1 + erf
2
dcz
where ηcz = 1010 cm2 s−1 , rcz = 0.69 R , and dcz = 0.07 R .
Since diffusivity is now a state variable, small errors can
lead to negative values of diffusivity, which in turn lead to
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The Astrophysical Journal Letters, 727:L23 (6pp), 2011 January 20
Effective Diffusivity
Muñoz-Jaramillo, Nandy, & Martens
Toroidal Field
Poloidal Field
Figure 3. Snapshots of the effective diffusivity and the magnetic field over half a dynamo cycle (a sunspot cycle). For the poloidal field, a solid (dashed) line corresponds
to clockwise (counterclockwise) poloidal field lines. Each row is advanced in time by a sixth of the dynamo cycle (a third of the sunspot cycle), i.e., from top to bottom
t = 0, τ/6, τ/3, and τ/2. As expected, the turbulent diffusivity is strongly depressed by the magnetic field (especially by the toroidal component). This reduces the
diffusive timescale to a point where the magnetic cycle becomes viable and sustainable.
(A color version of this figure is available in the online journal.)
intact, in order to compare their solutions with those of the
dynamically quenched simulation. We find that the geometric
average (p → 0; also known as logarithmic average) shown in
Figures 4(a) and (b) as solid lines captures best the essence of the
diffusive transport by striking a balance between high- and loworder values of the diffusivity. Interestingly, this average can
be accurately described as a double-step profile (Equation (5))
with the following parameters: η0 = 108 cm2 s−1 , η1 =
1.6 × 1011 cm2 s−1 , η2 = 3.25 × 1012 cm2 s−1 , r1 = 0.71 R ,
d1 = 0.017 R , r2 = 0.895 R , and d2 = 0.051 R (see circles
in Figure 4(b)).
In order to compare the general properties of both simulations we cast the results in the shape of synoptic maps (also
known as butterfly diagrams) as can be seen in Figure 5. The results obtained using the MLT estimate and diffusivity quenching
(Figure 5(a)) and the results obtained using a conventional kinematic simulation with the geometric average fit (Figure 5(b))
are remarkably similar given the different nature of the two
where p controls the relative importance given to high orders
of magnitude (p > 0) and low orders of magnitude (p < 0) of
the diffusivity. From this generalized mean one can obtain the
most commonly used averages: p → ∞ yields the maximum
value, p = 1 the algorithmic average, p → 0 the geometric
average, p = −1 the harmonic average, and p → −∞ the
minimum value. The results of calculating these averages are
shown in Figure 4(a). Note that one can also obtain temporal
averages in which the effective diffusivity depends both on
radius and latitude. However, these averages cannot be related
to the commonly used kinematic profiles which are exclusively
radial. Because of this, we leave the analysis and study of such
averages for future work.
6.2. Comparison with Kinematic Dynamo Simulations
Once we calculate the spatiotemporal averages of the effective
diffusivity we obtain radial diffusivity profiles that can be used
in kinematic dynamo simulations, leaving all other ingredients
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Muñoz-Jaramillo, Nandy, & Martens
14
14
10
10
M −>Max
inf
M1−>Arit
13
10
13
10
M0−>Geom
M−1−>Harm
M−inf−>Min
12
12
10
η (cm2 /s )
η (cm2 /s )
10
11
10
10
10
10
10
9
9
10
MLT Estimate
Geometric average
Double step Fit
10
8
10
11
10
8
0.6
0.7
0.8
r/Rs
0.9
10
1
0.6
0.7
0.8
r/Rs
0.9
1
(b )
(a)
Figure 4. Spatiotemporal averages of the effective diffusivity (a). We find that the geometric time average (b) captures the essence of diffusivity quenching the best.
(A color version of this figure is available in the online journal.)
(meridional circulation and differential rotation). This feedback
can arise due to direct Lorenz force feedback on the flow fields
(Rempel 2006a) and/or magnetic quenching of the turbulent
viscosity (which in turn can affect angular momentum transfer; Rempel 2006b). Direct Lorenz force feedback has not been
found to have a strong impact on the interaction of the magnetic
field with meridional flow, but it does affect toroidal field generation through the reduction of rotational shear (Rempel 2006a);
on the other hand turbulent viscosity quenching has been found
to alter the meridional flow pattern (Rempel 2006b). Ultimately,
both diffusivity quenching and magnetic feedback on the global
flow patterns need to be taken into account in order to have a
complete picture. Nevertheless, this work defines a framework
which can be used to find a physically meaningful diffusivity
profile based upon theory and models of the solar interior rather
than through heuristic approaches.
Surface Radial Field and AR emergence
(a)
6.3. Comparison with Solar Cycle Observations
Ultimately, the goal of dynamo models is to understand
the solar magnetic cycle and reproduce and predict its main
characteristics. It is therefore important to compare our results
with solar observations. It is clear that the solutions are not
as successful in reproducing solar characteristics as those of
kinematic dynamo simulations whose parameters have been
finely tuned: cycle period of seven years instead of eleven,
broad wings, and a slight mismatch between the observed
and simulated phases of the peak polar field. Guerrero et al.
(2009) also find this issue in their simulations, pointing out that
the problem may reside in the functional form of diffusivity
quenching. Unfortunately, recent MHD simulations attempting
to find the validity of the functional form shown in Equation (4)
have obtained inconclusive results (Käpylä & Brandenburg
2009). It is clear that additional work is required in order to
fully reconcile diffusivity quenching with cycle observations.
However, the solutions we obtain are reasonably accurate, given
the fact that we have refrained from doing any fine tuning,
limiting ourselves to fundamental theories and models.
(b)
Figure 5. Synoptic maps (butterfly diagrams) showing the time evolution of the
magnetic field in a simulation using the mixing-length theory (MLT) estimate
and diffusivity quenching (a), and a kinematic simulation using the geometric
spatiotemporal average of the dynamically quenched diffusivity (b). They are
obtained by combining the surface radial field and active region emergence
pattern. For diffuse color, red (blue) corresponds to positive (negative) radial
field at the surface. Each red (blue) dot corresponds to an active region emergence
whose leading polarity has positive (negative) flux. In both sets of simulations
we restrict the eruption of active regions in latitude as explained in Section 4. We
can see that a kinematic dynamo simulation using the geometric spatiotemporal
average can capture the most important features of the magnetic cycle produced
by the simulation using the MLT estimate and diffusivity quenching (period,
amplitude, and phase).
(A color version of this figure is available in the online journal.)
simulations: the shape of the solutions differs mainly in the AR
emergence pattern. However, the general properties of the cycle
(amplitude, period, and phase) are successfully captured by the
geometric average and are essentially the same. It is important to
mention that in these simulations we have not taken into account
the feedback of the magnetic field on the global velocity flows
7. CONCLUSIONS
In summary, we have shown that coupling magnetic
quenching of turbulent diffusivity with the estimated profile
from MLT allows kinematic dynamo simulations to produce
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The Astrophysical Journal Letters, 727:L23 (6pp), 2011 January 20
Muñoz-Jaramillo, Nandy, & Martens
solar-like magnetic cycles, which was not achieved before.
Therefore, we have reconciled MLT estimates of turbulent diffusivity with kinematic dynamo models of the solar cycle. Furthermore, we have demonstrated that kinematic simulations using
a prescribed diffusivity profile based on the geometric average
of the dynamically quenched turbulent diffusivity are able to
reproduce the most important cycle characteristics (amplitude,
period, and phase) of the non-kinematic simulations. Incidentally, this radial profile can be described by a double-step profile,
which has been used extensively in recent solar dynamo simulations. Although using this profile does not yield solutions as
close to solar observations as do finely tuned profiles and the
combined effect of diffusivity quenching and magnetic feedback
on the global flows still needs to be explored, our approach paves
the way for reconciling MLT with kinematic dynamo models.
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We are grateful to Dana Longcope and Paul Charbonneau
for vital discussions during the development of this work. We
thank Jørgen Christensen-Dalsgaard for providing us with Solar
Model S data and useful counsel regarding its use. We also
thank anonymous referees for very thorough reviews which led
to an overall improvement of this Letter. The computations
required for this work were performed using the resources
of Montana State University and the Harvard-Smithsonian
Center for Astrophysics. We thank Keiji Yoshimura at MSU,
and Alisdair Davey and Henry (Trae) Winter at the CfA for
much appreciated technical support. This research was funded
by NASA Living With a Star grant NNG05GE47G and has
made extensive use of NASA’s Astrophysics Data System. D.N.
acknowledges support from the Government of India through
the Ramanujan Fellowship.
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